$$\sin x
ps:
By differentiation, monotonicity and Taylor formula, all are wrong, because (sinx)′=cosx must use limx→0sinxx=1, and this formula must use sinx<x. This is vicious circle.
If we use Taylor series of sinx to define sinx, strictly prove $\sin x
Answer
We can define sinx as power series. Applying the knowledge of power series, obtain the derivative of sinx, and then we will easy prove the inequality. Concluding geometry of sinx, please refer to this.
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